Question

What is `int (sin x)/(cos^3 x) dx`?

Answer 1

`=1/2 sec^2(x)+C` or `=1/2 tan^2(x) +C`

Explanation:

`int sin(x)/cos^3(x) dx`

Let us use the substitution method.

We can see that the derivative of `cos(x)` is `-sin(x)`

Let `cos(x)= u`
Differentiating with respect to `x` we get
`-sin(x)dx = du`
`sin(x)dx = -du`

Our integral becomes

`int (-du)/u^3`
`=-int u^-3 du`

`=-(u^(-3+1)/(-3+1))+C`

`=-u^-2/-2+C`
`=1/(2u^2)+C`

Substituting back

`=1/(2cos^2(x))+C`

We can also write this as
`=1/2 sec^2(x)+C`

Note this can also be written as `1/2 tan^2(x) +C`
If you are wondering why? It is nothing but replacing `sec^2(x)` as `1+tan^2(x)`. Let me show how.

`=1/2(1+tan^2)x+C`
`=1/2+1/2(tan^2(x)+C`
`=1/2tan^2(x)+C` since `C` is a constant and `1/2+C` would be a constant too. We can write it as `C` itself.

For integration involving trigonometric functions there are usually more than one way of giving an answer so just watch out.